Ronald O. Fulp

Geometric Prequantization on the Spin Bundle Based on N-Symplectic Geometry: The Dirac Equation

AbstractWe present preliminary results for a prequantization procedure that leads in a natural way to the Dirac equation. The starting point is the recently introducedn-symplectic geometry on the bundle of linear framesLM of ann-dimensional manifoldM in which the ℝn-valued soldering 1-formθ onLM plays the role of then-symplectic potential. On a 4-dimensional spacetime manifold we consider the tensorial ℝ4⊗ℝ4valued functionĝ onLM determined by the spacetime metric tensor g as the Hamiltonian for free observers and determine the associated ℝ4-valued Hamiltonian vector field

Generalized Symplectic Geometry as a Covering Theory for the Hamiltonian Theories of Classical Particles and Fields

\hat X_{\hat g} = \hat X_{\hat g}^i \otimes r_i

Canonical transformations of local functionals and sh-Lie structures

, Integration” of theXĝi yields the dynamics of free observers on spacetime, namely parallel transport of linear frames along spacetime geodesies. In order to obtain a vector field on the spin bundleSM which is a lift of

Affine geometries defined by fiber preserving diffeomorphisms

\hat X_{\hat g}

Algebra Structures on Hom(C, L)

and which is induced by a vector field